The stress and strain in the vessel wall are important determinants of vascular physiology and patho-physiology. Vessels are constrained radially by the surrounding tissue. The hypothesis in this work is that the surrounding tissue takes up a considerable portion of the intravascular pressure and significantly reduces the wall strain and stress. Ten swine of either sex were used to test this hypothesis. An impedance catheter was inserted into the carotid or femoral artery, and after mechanical preconditioning pressure-cross sectional area relations were obtained with the surrounding tissue intact and dissected away (untethered), respectively. The radial constraint of the surrounding tissue was quantified as an effective perivascular pressure on the outer surface of the vessel, which was estimated as 50% or more of the intravascular pressure. For carotid arteries at 100 mmHg pressure, the circumferential wall stretch ratio in intact state was about 20% lower than the untethered state, and the average circumferential stress was reduced by about 70%. For femoral arteries, the reductions were about 15% and 70%, respectively. These experimental data support the proposed hypothesis, and suggest that in vitro and in vivo measurements of the mechanical properties of vessels must be interpreted with consideration of the constraint of the surrounding tissue.